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Fibonacci Hyperbolic Functions

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Let

psi=1+phi
(1)
=1/2(3+sqrt(5))
(2)
=2.618033...
(3)

(OEIS A104457), where phi is the golden ratio, and

alpha=lnphi
(4)
=0.4812118
(5)

(OEIS A002390).

FibonacciSinh

Define the Fibonacci hyperbolic sine by

sFh(x)=(psi^x-psi^(-x))/(sqrt(5))
(6)
=(phi^(2x)-phi^(-2x))/(sqrt(5))
(7)
=2/(sqrt(5))sinh(2xalpha).
(8)

The function satisfies

sFh(-x)=-sFh(x),
(9)

and for n in Z,

sFh(n)=F_(2n),
(10)

where F_n is a Fibonacci number. For n=1, 2, ..., the values are therefore 1, 3, 8, 21, 55, ... (OEIS A001906).

FibonacciCosh

Define the Fibonacci hyperbolic cosine by

cFh(x)=(psi^(x+1/2)+psi^(-(x+1/2)))/(sqrt(5))
(11)
=(phi^((2x+1))+phi^(-(2x+1)))/(sqrt(5))
(12)
=2/(sqrt(5))cosh[(2x+1)alpha].
(13)

This function satisfies

cFh(-x)=cFh(x-1),
(14)

and for n in Z,

cFh(n)=F_(2n+1),
(15)

where F_n is a Fibonacci number. For n=1, 2, ..., the values are therefore 2, 5, 13, 34, 89, ... (OEIS A001519).

FibonacciTanh

Similarly, the Fibonacci hyperbolic tangent is defined by

tFh(x)=(sFh(x))/(cFh(x)),
(16)

and for x in Z,

tFh(n)=(F_(2n))/(F_(2n+1)).
(17)

For n=1, 2, ..., the values are therefore 1/2, 3/5, 8/13, 21/34, 55/89, ... (OEIS A001906 and A001519).

See also

Fibonacci Number

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References

Sloane, N. J. A. Sequences A001519/M1439, A001906/M2741, A002390/M3318, and A104457 in "The On-Line Encyclopedia of Integer Sequences."Stakhov, A. and Tkachenko, I. "Hyperbolic Fibonacci Trigonometry." Dokl. Akad. Nauk Ukrainy, No. 7, 9-14, 1993.Trzaska, Z. W. "On Fibonacci Hyperbolic Trigonometry and Modified Numerical Triangles." Fib. Quart. 34, 129-138, 1996.

Referenced on Wolfram|Alpha

Fibonacci Hyperbolic Functions

Cite this as:

Weisstein, Eric W. "Fibonacci Hyperbolic Functions." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FibonacciHyperbolicFunctions.html

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